Question

question 10 · 1 point
which functions have graphs with no horizontal asymptote and no slant asymptote? select all that apply.
select all that apply:
□ $f(x) = \frac{x^3}{x - 3}$
□ $f(x) = \frac{3x}{5x + 4}$
□ $f(x) = \frac{8x^4}{x + 2}$
□ $f(x) = \frac{x^4 + 4x}{x^3 - 3}$
□ $f(x) = \frac{2x^3}{x^3 - 3}$

Explanation:

Step1: Check degree difference for each function

For rational functions $f(x)=\frac{P(x)}{Q(x)}$:

• Horizontal asymptote (HA): $\deg(P)\leq\deg(Q)$
• Slant asymptote (SA): $\deg(P)=\deg(Q)+1$
• No HA/SA: $\deg(P)\geq\deg(Q)+2$

Step2: Analyze each option

1. $f(x)=\frac{x^3}{x-3}$: $\deg(P)=3$, $\deg(Q)=1$ → $3-1=2$ → No HA/SA
2. $f(x)=\frac{3x}{|2x|+4}$: $\deg(P)=1$, $\deg(Q)=1$ → HA exists
3. $f(x)=\frac{8x^3}{x+2}$: $\deg(P)=3$, $\deg(Q)=1$ → $3-1=2$ → No HA/SA
4. $f(x)=\frac{x^4+4x}{x^3-3}$: $\deg(P)=4$, $\deg(Q)=3$ → $4-3=1$ → SA exists
5. $f(x)=\frac{2x^5}{x^3-3}$: $\deg(P)=5$, $\deg(Q)=3$ → $5-3=2$ → No HA/SA

Answer:

$f(x)=\frac{x^3}{x-3}$, $f(x)=\frac{8x^3}{x+2}$, $f(x)=\frac{2x^5}{x^3-3}$